Can a meniscus tear heal on its own?

Yes, it can, BUT it- Is rather unlikely, as they generally don't, unless the tear is a minor one. I don't think they heal but they certainly become symptom-free. Surgery can help certain types of the tears by repairing them which are dependent on the specific site they occur in. Most are however not amenable to repair.
Rarely. Much of the menisci do not have a blood supply, thus cannot heal. The only ones that have a chance of healing are the tears on the periphery of the meniscus, where there is a decent blood supply.